ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS

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1 ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS KAZUO AKUTAGAWA, LUIS A. FLORIT, AND JIMMY PETEAN Abstract. For a closed Riemannian manifold M m, g) of constant positive scalar curvature and any other closed Riemannian manifold N n, h), we show that the limit of the Yamabe constants of the Riemannian products M N, g + rh) as r goes to infinity is equal to the Yamabe constant of M m R n, [g + g E ]) and is strictly less than the Yamabe invariant of S provided n 2. We then consider the minimum of the Yamabe functional restricted to functions of the second variable and we compute the limit in terms of the best constants of the Gagliardo-Nirenberg inequalities. 1. Introduction Let M m be a closed smooth manifold of dimension m and denote by [g] the conformal class of a Riemannian metric g on M. The Yamabe constant Y M, [g]) of [g] is the infimum of the normalized total scalar curvature functional restricted to [g]: Y M, [g]) = inf h [g] inf Q g f) := f W 1,2 M) f 0 s M hdµ h VolM, h) m 2 m where s h is the scalar curvature of h and dµ h its volume element. Of course, the Yamabe constant can be expressed in terms of functions in the Sobolev space W 1,2 M) by writing h = f 4 m 2 g if f C + M)): am f 2 M g + s g f 2) dµ g Y M, [g]) = f 2, p m inf f W 1,2 M) f 0 where a m = 4m 1) and p m 2 m = 2m. m 2 Functions realizing the infimum are called Yamabe minimizers and the corresponding metrics are called Yamabe metrics and have constant scalar curvature). They always exist by a fundamental theorem obtained in several steps by H. Yamabe, N. Trudinger, T. Aubin and R. Schoen [29, 27, 3, 21]. Then one defines the Yamabe invariant of M, Y M), as the supremum of the Yamabe constants of all conformal classes of metrics on M [22, 12] cf. [11]). This invariant is always finite since for any conformal class [g], Y M m, [g]) Y S m, [g S m]) = mm 1)VolS m, g S m) 2/m, where g S m is the round metric on S m of constant sectional curvature 1. We will denote Y m := Y S m ) = Y S m, [g S m]). K. Akutagawa is supported in part by the Grants-in-Aid for Scientific Research C), Japan Society for the Promotion of Science, No L. A. Florit is partially supported by CNPq-FAPERJ, Brazil. J. Petean is supported by grant E of CONACYT and CAPES-Brazil. 1,

2 2 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN The nature of the problem of computing or estimating the Yamabe constant of a conformal class, and therefore the Yamabe invariant of the manifold, depends strongly on whether the constant is positive or non-positive. If Y M, [g]) 0 then Y M, [g]) inf M s g )VolM, g) m, 2 as was first pointed out by O. Kobayashi in [12]. This allows for instance to study the behavior of the Yamabe invariant under surgery [18]) and so to obtain some understanding of the invariant in the non-positive case; [19, 20, 6]. Such an estimate does not exist in the positive case. In particular there might exist unit volume Riemannian metrics on M m of constant scalar curvature greater than Y m. A typical example of this situation comes from Riemannian products: if M m, g) and N n, h) are unit volume Riemannian manifolds of constant scalar curvature and s g > 0 then, for small δ > 0, δ n g +δ m h has volume one and scalar curvature greater than Y. It is the main purpose of this article to study the Yamabe constants of such Riemannian products. There is one well understood example in this direction worked out by R. Schoen [22] and O. Kobayashi [11, 12]: for any r > 0 all solutions of the Yamabe equation on S n 1 S 1, g +rg 1) depend only on the S n 1 S1 -variable and one can understand the S solutions of the resulting ordinary differential equation. Following this lead, we will consider Riemannian products δ n g + δ m h on M N and look for solutions of the Yamabe equations which depend only on the second variable. Let Y N M N, g + h) := inf Q g+h f). f W 1,2 N) f 0 One can see that the infimum is realized and that if f is such a minimizer then 4 f 2 g + h) has constant scalar curvature we will go over this on Section 2). We remark that, contrarily to the Yamabe constant Y M N, [g + h]), this constant Y N M N, g + h) is not a conformal invariant, but merely a scale invariant. Our first result says in particular that the limit of the Yamabe constant of the products above exists: Theorem 1.1. Let M m, g) be a closed Riemannian m-manifold m 2) of positive scalar curvature not necessarily constant) and N n, h) any closed Riemannian n- manifold. Then, and lim Y M N, [g + rh]) = Y M rր Rn, [g + g E ]) := lim Y NM N, g + rh) = Y R nm R n, g + g E ) := rր where, g E stands for the Euclidean metric on R n. inf Q f Cc g+ge f) > 0, M Rn ) f 0 inf Q f Cc g+ge f) > 0, Rn ) f 0 Remark 1.2. The fact that Y S n 1 S 1 ) = Y S n ) for n 3) was first proved by O. Kobayashi [11] and R. Schoen [22], by analysis of the behavior of constant scalar curvature metrics on R n {0}, g E ). Another proof was given by also O. Kobayashi [12],

3 ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS 3 by using an argument in the proof of the celebrated Kobayashi s inequality [12, Theorem 2]. The above theorem gives the third proof since Y S n 1 S 1 ) Y S n 1 R 1, [g S n 1 + g E ]) = Y Sn, [g S n]) = Y S n ) ). On the constant Y M R n, [g + g E ]), we also obtain: Theorem 1.3. Let M m, g) be a closed Riemannian m-manifold m 2) of positive scalar curvature not necessarily constant). Assume that n 2. Then, Y M R n, [g + g E ]) < Y. Recall that the best n-dimensional Sobolev constant is the smallest positive number σ n such that for any smooth compactly supported function f on R n, f 2 p n σ n f 2 2. Due to the conformal invariance of the Yamabe constant, one can use the stereographic projection to translate the Yamabe functional from the round sphere to the Euclidean space to obtain: σ n = a n Y n. In a similar fashion we will see that when studying the limits above a fundamental role is played by the best constant in the Gagliardo-Nirenberg inequalities: namely, we will call σ m,n the smallest positive number such that for any f W 1,2 R n ), Or what is equivalent: f 2 p σ m,n f 2n 2 f 2m σ m,n = inf f W 1,2 R n ) f 0 f 2n 2. 2 f 2m 2 f 2 p The constant σ m,n is of classical interest in the study of partial differential equations and has been computed numerically, although it is not known any explicit expression for the constant or for the minimizing function which is known to be radial and decreasing); [8, 9, 17, 13, 16, 28]. We will prove: Theorem 1.4. Let M m, g) be a closed smooth unit volume Riemannian manifold of constant positive scalar curvature s g. Then Y R nm R n, g + g E ) = where Cm, n) = a ) n m + n)n n m m. It is clear that 1 Cm, n) s m g σ m,n, Y M R n, [g + g E ]) Y R nm R n, g + g E ) and it seems that equality should hold under certain hypothesis. It certainly cannot always be the case since Y R nm R n, g + g E ) > Y if s g is big enough..

4 4 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN It is then natural to ask the following Question: Is it true that Y R nm R n, g + g E ) = Y M R n, [g + g E ]) if g is a Yamabe metric? As we mentioned before, the constants σ m,n can be explicitly computed numerically. Using these computations, we apply Theorem 1.4 to the case when M, g) and N, h) are round spheres. These are particularly interesting cases because of Schoen and Kobayashi s argument mentioned above and because S n S m is obtained by performing surgery on S, and therefore if the surgery theorem in [18] were true in the positive case we should have Y S m S n ) = Y. Set Y S nsm S n ) := lim rր Y S ns m S n, g S m + rg S n) = Y R ns m R n, g S m + g E ). We give the corresponding values for all m, n 2 with m + n 9. m n σm,n 1 YS nsm S n ) Y It should be the case that Y S nsm S n ) < Y n+m for all values m, n 2. Hence, this gives a proof of Theorem 1.3 for M m, g) = S m, g S m), namely that of Y S m R n, [g S m + g E ]) < Y

5 ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS 5 when m, n 2, m + n 9. Moreover, the above method gives a numerical estimate from above for the constant Y S m R n, [g S m + g E ]). Note also that in the 4 dimensional case the Yamabe invariant of CP 2 is realized by the conformal class of the Fubini-Study metric g FS [14, 10]), giving Y CP 2 ) = Y CP, [g FS ]) = 12 2π = and since Einstein metrics are always Yamabe metrics we have that Y S 2 S 2, [g S 2 + g S 2]) = 16π = In the next section, we will review some known results on Yamabe constants, point out a few observations and fix some notation. In Section 3, we will recall Schoen and Kobayashi s discussion of the solutions of the Yamabe equation on S n 1 S 1, g S n 1 rg S 1). We will prove Theorem 1.1 and Theorem 1.3 in Section 4 and Theorem 1.4 in Section 5. Finally, a procedure to compute numerically these Yamabe constants is given in the last section. Acknowledgements: The authors would like to thank Professors Manuel del Pino and Jean Dolbeault for valuable observations on the Gagliardo-Niremberg inequalities and Fernando Coda for numerous conversations on the subject. The first author would like to express his gratitude to Professors Boris Botvinnik and Osamu Kobayashi for many helpful discussions. The third author would like to express his gratitude to IMPA where this work was carried on. 2. Preliminaries Let X k, g) be a closed smooth k-dimensional Riemannian manifold. Recall that s g is the scalar curvature of g, dµ g its volume element and p = p k = 2k k 2 and a = a k = 4k 1) k 2. Consider the Sobolev space W 1,2 X) and the Yamabe functional defined by ak f 2 X g f Q g f) := dµ g + s g f 2) dµ g f 2. p We say that f is a Yamabe minimizer for g) if it realizes the minimum of the Yamabe functional. In this case f 4/k 2) g has constant scalar curvature and the Yamabe constant of the conformal class of g is then equal to Q g f). In this paper we consider a unit volume Riemannian product M m N n, g+h). We assume that the scalar curvature of both g and h are constant and try to understand the Yamabe constant of the conformal class of the product metric. This is of no interest if s g + s h is negative, since in this case we have uniqueness of the Yamabe metric. The situation we want to address is when s g + s h is positive and bigger than Y, the Yamabe invariant of the round sphere S. In this case there must exist

6 6 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN a non-constant Yamabe function, and so another metric of constant scalar curvature in the same conformal class. To compute the Yamabe constant is a very difficult problem and so it is to understand the Yamabe minimizer. We will then restrict ourselves to functions which depend only on one of the variables, that is, positive smooth functions f : N n R of one of the factors in the Riemannian product M m N n. The scalar curvature of 4 f 2 g + h) is given by s g+h = f 1 p a h f + s g+h f). We then introduce the following definition: Definition 2.1. Given a Riemannian product M N, g + h) of constant scalar curvature manifolds, the N Yamabe constant of M N, g + h) is the infimum of the g + h) Yamabe functional restricted to W 1,2 N). We will denote this constant by Y N M N, g + h). To study critical points of the g + h) Yamabe functional restricted to W 1,2 N), let ϕ, f : N R be smooth functions. A well-known computation gives that d Qf + tϕ) ) 2VolM, g) ) 0) = dt f 2 a h f +s g+h f f p+2 p Qf)f p 1 ϕ dµ h, p N where p = p and Q ) = Q g+h ). Therefore the critical points of the Yamabe functional restricted to W 1,2 N) are precisely the functions f such that the conformal metric f 4/ 2) g + h) has constant scalar curvature s = f p+2 p Qf). The next point is that the infimum of the Yamabe functional restricted to W 1,2 N) is always achieved. This is a simple fact, it is essentially the subcritical case of the Yamabe problem, but we sketch its proof since we have not seen it in the literature. Proposition 2.2. Let M m, g) and N n, h) be closed Riemannian manifolds of constant scalar curvature. Then, there exists a positive smooth function f on N n which minimizes the Yamabe quotient among all functions on N n. Proof. Without loss of generality, we may assume that VolM, g) = 1. Let {u i } be a sequence of non-negative functions on N such that Q g+h u i ) Y N M N, g + h). We can assume that u i pk = 1, where k = m + n. Then, u i 2 2,1 = ui 2 h + ) u2 i dµh N = 1 s g+h ) Q g+h u i ) + 1 u 2 a k a i dµ h k N Y NM N, g + h) + 1 a k + K u i 2 2,

7 ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS 7 for some K > 0, that is bounded independently of i since u i 2 2 u i 2 p k VolN, h) 2/k = VolN, h) 2/k by Hölder s inequality. Since 1 p k > n, by the Rellich-Kondrakov Theorem the inclusion W 1,2 N) L p k N) is a compact operator. Therefore, there exists a subsequence of the {u i } which converges weakly in W 1,2 N) and strongly in L p k N) to a function u W 1,2 N) with u pk = 1. Now, by the weak convergence in W 1,2 N), we have u 2 = lim u, u 2 i dµ h, i and therefore u 2 2 N lim sup ui 2. 2 i And since by strong convergence in L 2 s g+h u 2 dµ h = lim s g+h u N i N 2 idµ h, we have that Q g+h u) lim i Q g+h u i ), and hence Q g+h u) = Y N M N, g +h). It then follows from elliptic regularity that u has to be strictly positive and smooth. Remark 2.3. Note that, for a given Riemannian product of constant scalar curvature, we have three associated Yamabe constants each producing a constant scalar curvature metric. The three are equal if the original product is a Yamabe metric. 3. Reviewing the circle Schoen [22] cf. Kobayashi [11]) gave a fairly complete study of the solutions of the Yamabe equation for the manifolds S n 1 S 1, g S n 1 + rg S 1), where n 2 and r is a positive constant. He points out that due to the conformal invariance and a theorem of Caffarelli Gidas Spruck [7] all solutions are functions of S 1. Moreover, he writes down the Yamabe equation for a function of S 1. Moving to the Riemannian universal covering S n 1 R, g S n 1 +dt 2 ), one has to deal with the ordinary differential equation: d 2 u dt n nn 2) 2)2 u + u n+2)/n 2) = 0, 4 and look for solutions which are 2πr periodic. Note that exactly the same equation shows up if we consider a Riemannian product M S 1, g+rg S 1), where M is n 1)- dimensional and the scalar curvature of g is n 1)n 2). In this way, one can understand all constant scalar curvature metrics which are conformal by a function of S 1 to M S 1, g+rg S 1); the solutions are the same as those for S n 1 discussed in [22] and [11]. So for r close to 1 there is going to be only one solution, and as r increases the number of solutions will increase. If VolM, g) = V n 1 := VolS n 1, g S ), then n 1

8 8 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN the S 1 -Yamabe constant of the product will be less than Y n for all r and will approach Y n as r goes to infinity. Example 3.1. If VolM, g) > V n 1 in the discussion above, the number of solutions will still be the same, but as r becomes big the S 1 Yamabe constants of the product will be bigger than Y n. In particular, the S 1 Yamabe constant will be greater than the Yamabe constant. 4. Proofs of Theorem 1.1 and Theorem 1.3 Proof of Theorem 1.1. To simplify the notation, we set g r := g +rh on M m N n and ḡ := g + g E on M m R n. We may also assume that VolM, g) = 1. First, we show the following Y M R n, [ḡ]) > 0. Note that M R n, ḡ) is a complete Riemannian manifold with strictly positive injective radius and bounded sectional curvature. Under these conditions, the Sobolev embedding W 1,2 M R n ) L p M R n ) holds cf. [4, Theorem 2.21]), that is, there exists constant K 1 > 0 such that f 2 p K 1 f 2 2,1 for f W 1,2 M R n ), where p = p := 2). This and the positivity of the scalar curvature of M, g) 2 imply that ) 2/p f p K 1 dµḡ a f 2 ḡ M R α + s gf 2) dµḡ n M R n for f W 1,2 M R n ), where α := min { a, min s } g > 0, and hence M Y M R n, [ḡ]) α K 1 > 0. We also have Y R nm R n, ḡ) Y M R n, [ḡ]) > 0. Second, we prove the following 1) lim inf rր and Y M N, [g r]) Y M R n, [ḡ]) 2) lim inf rր Y NM N, g r ) Y R nm R n, ḡ). Pick any ε > 0. There exist a small constant δ > 0 and finite points {q 1,, q l } N such that {U k := exp h q k B h δ 0) ) } l k=1 is an open covering of N and that, on each U k with respect to h-normal coordinates x = x 1,, x n ) at q k, 1+ε) 1 δ ij dx i dx j h ij dx i dx j 1+ε)δ ij dx i dx j, 1+ε) 1 dx dµ h 1+ε)dx.

9 ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS 9 Here, exp h q k : B h δ 0) := {v T q k N v h < δ} U k and dx := dx 1 dx n denote respectively the h-exponential map at q k and the Euclidean volume form. Then note that, for any r > 1, on each U i with respect to r 2 h)-normal coordinates y = y 1,, y n ) at q k, 1 + ε) 1 δ ij dy i dy j r 2 h) ij dy i dy j 1 + ε)δ ij dy i dy j, 1 + ε) 1 dy dµ r 2 h 1 + ε)dy. We also note that there exists a constant K 2 > 0 such that diamu k, r 2 h) K 2 r k = 1,, l) for any r 1. Let {η k = χ 2 k }l k=1 be a partition of unity subordinate to the covering {U k } l k=1 and K 3 > 0 a positive constant independent of r 1 such that and hence χ k h K 3 k = 1,, l), χ k r 2 h K 3 k = 1,, l). r With the above understandings, for any r > 1 and ϕ C M N), we estimate the L p -norm of ϕ with respect to g r as follows: ϕ 2 p = ϕ2 p/2 = Σ k χ 2 k ϕ2 p/2 Σ k M U k χ k ϕ p dµ gr ) ) 2/p 1 + ε) 2/p Σ k M U k χ k ϕ p dµḡ) 2/p. Here, we identify U k = exp r2 h q k B r 2 h rδ 0)) N) with B rδ 0) := {y R n y < rδ} via the composition of the inverse ) exp r2 h 1 q k and the identification B r 2 h rδ 0) = B rδ 0). Set Y 0 := Y M R n, [ḡ]) and Y0 0 := Y R nm Rn, ḡ). We also note that, on each M U k M R n ), ) 2/p χ k ϕ p 1 ) dµḡ a χ k ϕ) M U k Y 0 M U 2 ḡdµḡ + s g χ 2 lϕ 2 dµḡ k M U k 1 + ) ε)2 a χ k ϕ) Y 0 M U 2 g r dµ gr + s g χ 2 k ϕ2 dµ gr k M U k 1 + ε)3 a χ 2 k Y ϕ 2 g r dµ gr + s g χ 2 k ϕ2 dµ gr 0 M U k M U k + a K ) ε 1 ) ϕ 2 dµ r 2 gr. M U k Here, it is important to note for the proof of 2) that, if ϕ C N), then we can replace Y 0 by Y 0 0.

10 10 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN Hence, we have ϕ 2 p 1 + ε) 3+2/p) Y 0 M N ) a ϕ 2gr + s g ϕ 2 dµ gr + la K ε 1 ) ϕ 2 dµ r 2 gr ). M N From the positivity of the scalar curvature s g, there exists a large constant r 0 = r 0 ε, min s g, N, h), m + n) > 1 such that M Therefore, we obtain ϕ 2 p la K ε 1 ) r ε)4+2/p) Y 0 M N min M s ) g ε. a ϕ 2gr + s g ϕ 2 ) dµ gr for any r r 0 and ϕ C M N). Again, if ϕ C N), we can replace Y 0 by Y 0 0. Then, this implies that Y M N, [g r ]) for any r r 0. And since ε > 0 is arbitrary, and Finally, we prove Y ε) 4+2/p), Y NM N, g r ) lim inf rր Y M N, [g r]) Y 0 lim inf rր Y NM N, g r ) Y 0 0. Y ε) 4+2/p) 3) lim sup Y M N, [g r ]) Y 0 rր and 4) lim sup Y N M N, g r ) Y0 0. rր Note that 5) lim ρր Y M B ρ 0), [ḡ]) = Y 0, and 6) lim ρր Y Bρ0)M B ρ 0), [ḡ]) = Y 0 0.

11 ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS 11 Take any small ε > 0 and large ρ > 0. Fix a point q N and set an h-normal open neighborhood U := expq h B h δ 0) ) of q for δ > 0. Here, we choose δ = δε, N, h)) > 0 sufficiently small satisfying the same conditions on U as those in the preceding argument. Let r 1 > 0 be a positive constant such that r 1 δ ρ. For each r r 1, we also use the r 2 h)-normal coordinates y = y 1,, y n ) at q on U and the identification B rδ 0) R n = ) U = exp r2 h q B r 2 h rδ 0)) N). With the above understandings, for any r r 1 and f Cc M B ρ 0)) Cc M B rδ 0)) ) = Cc M U), we obtain f 2 p = M B ρ0) 1 + ε) 2/p Y M N, [g r ]) 1 + ε)2+2/p) Y M N, [g r ]) ) 2/p f p dµḡ 1 + ε) 2/p M U M B ρ0) M U f p dµ gr ) 2/p a f 2 g r + s hr f 2) dµ gr a f 2 ḡ + s g + K ) ) 4 f 2 dµḡ, r where K 4 > 0 is a constant independent of r. Here, we also use the fact that Y M N, [g r ]) > 0 for any large r > 0. In order to prove 4) it is important to note that for f Cc B ρ0)), we obtain: f 2 p 1 + ε)2+2/p) Y N M N, g r ) M B ρ0) a f 2 ḡ + s g + K 4 r ) f 2 ) dµḡ. From the positivity of s g, there exists r 2 = r 2 ε, min M s g) > 0 such that s g + K 4 r 1 + ε)s g on M for any r r 2. Hence, for any r max{r 1, r 2 } and f Cc M B ρ 0)), we have f 2 p 1 + ε)3+2/p) a f 2 ḡ Y M N, [g r ]) + s gf 2) dµḡ. Therefore, Letting r ր, we then obtain M B ρ0) Y M N, [g r ]) 1 + ε) 3+2/p) Y M B ρ 0), [ḡ]). lim sup Y M N, [g r ]) 1 + ε) 3+2/p) Y M B ρ 0), [ḡ]). rր Letting also ρ ր and ε ց 0, lim sup Y M N, [g r ]) Y 0. rր And following the same steps we also prove 4). This completes the proof of Theorem 1.1.

12 12 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN We shall prove Theorem 1.3 by a series of lemmas. Throughout the rest of this section, we always assume the same conditions as in Theorem 1.3, that is, m, n 2 and s g > 0 on M m. To simplify the notation, we set ḡ := g + g E on M m R n. By the positivity of the scalar curvature s g > 0 of g and the condition that n 2, one can obtain the following. Lemma 4.1. M m R n, ḡ) is not locally conformally flat. Moreover, for any open set U of M m R n, the Weyl curvature Wḡ of ḡ never vanishes identically on U. When 6, similarly to Aubin s argument in [3], [15, Theorem B], Lemma 4.1 implies the existence of a family of nice test functions {ψ ε } ε>0 with Qḡψ ε ) < Y for sufficiently small ε > 0. Then we obtain the following see the proof of Proposition 6.6 in [2] for details). Lemma 4.2. Assume that m + n 6. Then, the assertion of Theorem 1.3 holds. Now we consider the remaining case when m + n = 4, 5 in Theorem 1.3. Let T n k := R n /2 k Z) n denote the quotient of R n for k = 0, 1, 2,. Let R n, g E ) T n k, h k) also denote the natural infinite Riemannian covering and P k : M m R n, ḡ) M m T n k, ḡ k) the induced Riemannian covering, where ḡ k := g + h k. Here note that each natural map M m T n k+1, ḡ k+1 ) M m T n k, ḡ k ) is also a finite Riemannian covering. Take any point q M m and fix it. Set p = q, 0) M m R n and p k := P k p) M m Tk n. Then, for each k, there exists a unique normalized positive Green s function G k with pole at p k for the conformal Laplacian Lḡk on M m Tk n, that is, G k C+ M m Tk n) {p k} ) with Lḡk G k = c δ pk on M m T n k and lim x pk G k x)distp k, x)) 2 = 1. Here, c > 0 and δ pk stand respectively for a specific universal constant and the Dirac δ-function at p k. The condition s g > 0 implies that the first eigenvalue λlḡ0 ) > 0 on M m T n 0. By the condition λl ḡ 0 ) > 0, there exists a unique normalized minimal positive Green s function G for Lḡ with pole at p M m R n. Moreover, there exist positive constants a, b, K with a < b, K 1 such that K 1 e b z Gx) Ke a z for x = y, z) M m {z R n z 1}. See [25], [2, Section 6] for details.) Let us consider the family of the scalar-flat, asymptotically flat manifolds X k, ḡ k,af ) := M m Tk n ) {p k }, G 4 ) 2 k ḡ k for k = 0, 1, 2, and the one X, ḡ AF ) := M m R n ) {p}, G 2ḡ) 4

13 ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS 13 with a singularity arising from the end of M m R n. We denote the mass of X, ḡ AF ) resp. X k, ḡ k,af )) by mḡ AF ) resp. mḡ k,af )). Then, a similar argument to the proof of [2, Theorem 6.13] implies lim mḡ k,af) = mḡ AF ). k Hence, from the positive mass theorem [23, 24, 22], we obtain the following. Lemma 4.3. Let X, ḡ AF ) be the scalar-flat, asymptotically flat manifolds with a singularity and dim X = 4, 5, defined as above. Then, mḡ AF ) 0. Now we can complete the proof of Theorem 1.3. Proof of Theorem 1.3. From Lemma 4.2, we consider the case when = 4, 5. We will prove that mḡ AF ) > 0. Then, by a similar argument to the proofs of [21, Theorem 1] and [26, Chapter 5, Theorem 4.1], the positivity of the mass mḡ AF ) > 0 implies the desired assertion Y M m R n, [ḡ]) < Y. Choose any large constant L 0 > 0 and fix it. Set X 0 := M m {z R n z L0 } ) {p} X. Then, there exist harmonic coordinates near infinity x = x 1,, x ) on X 0, ḡ AF ) [5] cf. [2, Lemma 6.17]). Namely, x i ) are smooth functions on X 0 satisfying ḡaf x i x i = 0 on X 0, ν = 0 on X 0 and which give asymptotically flat coordinates near infinity of X 0, ḡ AF ). Here, ν is the outward unit normal vector field normal to X 0. Now we suppose that mḡ AF ) = 0. Note that, under this assumption, ḡ AF is Ricciflat on X 0. See [1, Lemma 3.1] for details cf. [21, Lemma 3], [2, Proposition 6.14]). We now apply the Bochner technique to complete the proof. The harmonicity of x i ) implies that {dx i } are harmonic 1-forms on X 0, ḡ AF ). From the Bochner formula for 1-forms {dx i } combined with the conditions that xi = 0 on X ν 0 and RicḡAF = 0 on X, we have that cf. [5, Theorem 4.4], [15, Proposition 10.2]) 1 mḡ AF ) = Vol S 1 1) ) dx i 2 dµḡaf. X 0 Then, by applying the mass zero condition mḡ AF ) = 0 in the above, we obtain that the 1-forms {dx i } are parallel on X 0 with respect to ḡ AF. Since the coframe {dx i } is orthonormal at infinity, then {dx i } is a parallel orthonormal coframe everywhere on X 0, ḡ AF ). This implies that the map x = x 1,, x ) : X 0, ḡ AF ) R, g E ) is a local isometry, and hence ḡ is locally conformally flat on X 0. This gives a contradiction to Lemma 4.1. Therefore, mḡ AF ) > 0. This completes the proof. i=1

14 14 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN 5. Gagliardo-Nirenberg inequalities and Yamabe constants In this section we estimate the behavior of arbitrary N Yamabe constants in terms of the best constants in the Gagliardo-Nirenberg interpolation inequalities. Let us define the m, n) Gagliardo Nirenberg functional as Lf) = L m,n f) := f 2n 2 f 2m 2 f 2 for f W 1,2 R n ) with f 0. p Remark 5.1. The map L is invariant under two operations. First, if c is any non-zero constant then Lcf) = Lf). Second, if λ > 0 is a constant and f λ x) = fλx) then and Therefore, Lf λ ) = Lf). f λ 2m 2 = λ mn 2m f 2, f λ 2 p = λ 2n p n+m f 2 p f λ 2n 2 = λ 2 n)n 2n f 2. Let us recall the following definition from the introduction: Definition 5.2. The m, n) Gagliardo Nirenberg constant is given by σ m,n := 1 inf L m,n f)). f W 1,2 R n ) f 0 Remark 5.3. The constant σ m,n has already been studied in the literature. It is the best constant for the classical interpolation inequality due to Gagliardo and Nirenberg that says that L m,n is bounded away from zero cf [8], [9] and [17]). In [28], it is shown that σ m,n is closely related to the global existence of the nonlinear Schrödinger equation. The author showed also that σ m,n is always attained by a positive function ψ W 1,2 R n ) C R n ), called a ground state solution, that should then satisfy the corresponding Euler Lagrange equation 7) u u + u q = 0, q = m + n + 2 m + n 2 the Euler-Lagrange equation for L of course involves coefficients depending on f 2, f p and f 2 : one can use the previous remark to normalize the equation as above). For any function f W 1,2 R n ) C R n ), one can consider the spherical symmetrization f of f, that is, the radial function f satisfying Vol ge {f > t} = Vol ge {f > t} for any positive t. It is a classical result that Lf ) < Lf) if f f. It follows that ψ should be radial and decreasing. Finally, in [13] it is proved the uniqueness of the positive radial solution of 7) under the assumption that it vanishes at infinity. The key point to our purposes is that these facts give a simple procedure to compute numerically all Gagliardo Nirenberg constants. We will continue this discussion on Section 6.

15 ON YAMABE CONSTANTS OF RIEMANNIAN PRODUCTS 15 We now relate the Gagliardo Nirenberg constants with the N Yamabe constant of limiting products. Proof of Theorem 1.4. First fix a function f Cc Rn ) and consider the family of functions f λ as in Remark 5.1. Let us consider the map R λ Fλ) := n a f λ 2 g + s g f 2 E λ) dµge. f λ 2 p If we set and A := B := then Remark 5.1 implies that Fλ) = λ 2m its minimum at λ 0 = nb ma R n a f 2 g E dµ ge f 2 p R n s g f 2 dµ ge f 2 p, A+λ 2n and that this minimum is Fλ 0 ) = A n m m n m B m n m + n) = s Recall that Lf) = Lf λ ). Therefore, B. We see that this map achieves g lim Y N nmm N n, g + rh) = Y R nm m R n, g + g E ) rր = inf f C c Rn ) f 0 = inf f C c Rn ) f 0 Cm, n)lf). R n a f 2 g E + s g f 2) dµ ge f 2 p s m g 6. Numerical computations Cm, n)lf) = s m g Cm, n) σ m,n. We describe now a procedure to determine all Gagliardo Nirenberg constants σ m,n numerically, and how we obtained our table in the introduction. Consider the solution h = h α : R 0 R of the initial value problem 8) h t) + n 1 h t) ht) + ht) +2 2 = 0, h0) = α > 0, h 0) = 0. t This equation corresponds to the critical points of the m, n) Gagliardo Nirenberg functional L m,n see Remarks 5.1, 5.3), when restricted to radial functions ux) = h x ), which is enough to consider because of symmetrization. By the existence result in [28] and its uniqueness finally proved in [13], there exists only one value α = α 0 = α 0 m, n) that gives a positive solution h α0 that vanishes at infinity, called the ground state. To find α 0 numerically, we use [13] where it is shown that, for

16 16 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN Figure 1: h α for α > α 0. Figure 2: h α for α < α 0. values α > α 0, the solution h α vanishes exactly once and then oscillates about 1 Figure 1), while, for values α < α 0, it is positive and oscillatory about the value 1 Figure 2). A key point is the uniqueness of the solution of 8) when the initial value condition h0) = α is replaced by the boundary condition ht 0 ) = 0, for t 0 0, + ], and the fact that this solution has h < 0 in 0, t 0 ]; see Lemmas 9, 11 and Theorem in [13]. Finally, by [28] we have that σm,n 1 = L m,nh α0 ). For example, we can then compute α 0 = α 0 2, 2) for the ground state initial value, and hence σ2,2 1 = L 2,2h α0 ) In fact, Figure 1 corresponds to m = n = 2 and α = 2.208, while Figure 2 to α = Of course, if one wants to avoid the numerical computation, one could give estimates for σ m,n by carefully choosing functions. For instance we can show that σ 1 2,2 < < 2π by considering the function h : R 0 R that linearly interpolates the following data: h0) = 1, h0.1) = , h0.2) = , h0.3) = , h0.7) = , h0.9) = , h1.15) = , h1.3) = 0.34, h1.5) = , h1.85) = , h2.2) = , h2.6) = , h3) = , h3.5) = , h3.9) = 0.016, h4.3) = , h5) = , h6) = , h7) = , h8) = , h9) = , and ht) = 0, for t 10. Now, define the radial function f : R 2 R given by fx) = h x ). A straightforward computation gives that σ 1 2,2 L 2,2 f) < And then Y S 2 S2 S 2 ) < π) < 8 6 π = Y 4. References [1] K. Akutagawa, Aubin s lemma for the Yamabe constants of infinite coverings and a positive mass theorem, preprint, [2] K. Akutagawa, B. Botvinnik, Yamabe metrics on cylindrical manifolds, Geom. Funct. Anal ),

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18 18 K. AKUTAGAWA, L. FLORIT, AND J. PETEAN Department of Mathematics, Tokyo University of Science, Noda , Japan. address: akutagawa IMPA, Est. Dona Castorina, 110, , Rio de Janeiro, BRAZIL. address: CIMAT, A.P. 402, 36000, Guanajuato. Gto., México. address: